Normal form decomposition for Gaussian-to-Gaussian superoperators

TL;DR

This paper characterizes linear maps that send Gaussian states to Gaussian states, revealing their structure and potential use in quantum state certification, including a decomposition for one-mode cases and partial results for multi-mode scenarios.

Contribution

It generalizes known results by characterizing Gaussian-to-Gaussian superoperators without requiring positivity, and provides a decomposition for one-mode maps, extending understanding of their structure.

Findings

Characterization of Gaussian-to-Gaussian superoperators via characteristic functions

Decomposition of one-mode superoperators into phase-space dilatation and Gaussian channels

Partial decomposition results for multi-mode superoperators under specific conditions

Abstract

In this paper we explore the set of linear maps sending the set of quantum Gaussian states into itself. These maps are in general not positive, a feature which can be exploited as a test to check whether a given quantum state belongs to the convex hull of Gaussian states (if one of the considered maps sends it into a non positive operator, the above state is certified not to belong to the set). Generalizing a result known to be valid under the assumption of complete positivity, we provide a characterization of these Gaussian-to-Gaussian (not necessarily positive) superoperators in terms of their action on the characteristic function of the inputs. For the special case of one-mode mappings we also show that any Gaussian-to-Gaussian superoperator can be expressed as a concatenation of a phase-space dilatation, followed by the action of a completely positive Gaussian channel, possibly composed with a transposition. While a similar decomposition is shown to fail in the multi-mode scenario, we prove that it still holds at least under the further hypothesis of homogeneous action on the covariance matrix.